Using symbolic computing to obtain Lagrange-Euler solutions for nonlinear progressive waves on a uniform current

Abstract

In this study, a symbolic implementation is introduced to perform the Lagrange-Euler transformation for the solutions of nonlinear progressive water waves on a uniform current over a finite depth. In the computation, the solutions in the Lagrangian description are obtained first and transformed subsequently to the corresponding solutions in the Eulerian description. To accomplish an automatically symbolic computation, operators for obtaining Taylor-Fourier coefficients are introduced to convert the hierarchical system of governing differential equations into a system of algebraic equations. The fifth-order Eulerian and Lagrangian solutions in the literature are extended to the seventh order by the proposed method. The correctness of the solution is checked by Richardson extrapolation to the limit. For efficient utilization in practical engineering applications, the seventh-order Eulerian solutions are implemented in C++ codes with accuracy improvements over the existing solutions demonstrated. Furthermore, this study can be considered a constructive demonstration of the equivalence between the Lagrangian and the Eulerian solutions. Some source codes are freely available online and can be used for further studies.